Appleton-Hartree equation

Appleton-Hartree equation

The Appleton-Hartree equation, sometimes also referred to as the Appleton-Lassen equation is a mathematical expression that describes the refractive index for electromagnetic wave propagation in a cold magnetized plasma. The Appleton-Hartree equation was developed independently by several different scientists, including Edward Victor Appleton, Douglas Hartree and K. Lassen.

Equation

Full Equation

The equation is typically given as follows [ Citation | last = Helliwell | first = Robert | date = 2006 | title = Whistlers and Related Ionospheric Phenomena | edition = 2nd | publication-place = Mineola, NY | publisher = Dover | pages = 23-24 ] :

:n^2 = 1 - frac{X}{1 - iZ - frac{frac{1}{2}Y^2sin^2 heta}{1 - X - iZ} pm frac{1}{1 - X - iZ}left(frac{1}{4}Y^4sin^4 heta + Y^2cos^2 hetaleft(1 - X - iZ ight)^2 ight)^{1/2

Definition of Terms

n = complex refractive index

i = sqrt{-1}

X = frac{omega_0^2}{omega^2}

Y = frac{omega_H}{omega}

Z = frac{ u}{omega}

u = electron collision frequency

omega = 2pi f

f = wave frequency

omega_0 = 2pi f_0 = sqrt{frac{Ne^2}{epsilon_0 m = electron plasma frequency

omega_H = 2pi f_H = frac{B_0 |e{m} = electron gyro frequency

epsilon_0 = permittivity of free space

mu_0 = permeability of free space

B_0 = ambient magnetic field strength

e = electron charge

m = electron mass

heta = angle between the ambient magnetic field vector and the wave vector

Modes of Propagation

The presence of the pm sign in the Appleton-Hartree equation gives two separate solutions for the refractive index [ Citation | last = Bittencourt | first = J.A. | date = 2004 | title = Fundamentals of Plasma Physics | edition = 3rd | publication-place = New York, NY | publisher = Springer-Verlag | pages = 419-429 ] . For propagation perpendicular to the magnetic field, i.e., kperp B_0, the '+' sign represents the "ordinary mode," and the '-' sign represents the "extraordinary mode." For propagation parallel to the magnetic field, i.e., kparallel B_0, the '+' sign represents a left-hand circularly polarized mode, and the '-' sign represents a right-hand circularly polarized mode. See the article on electromagnetic electron waves for more detail.

Reduced Forms

Propagation in a Collisionless Plasma

If the wave frequency of interest omega is much smaller than the electron collision frequency u, the plasma can be said to be "collisionless." That is, given the condition

u ll omega,

we have

Z = frac{ u}{omega} ll 1,

so we can neglect the Z terms in the equation. The Appleton-Hartree equation for a cold, collisionless plasma is therefore,

:n^2 = 1 - frac{X}{1 - frac{frac{1}{2}Y^2sin^2 heta}{1 - X} pm frac{1}{1 - X}left(frac{1}{4}Y^4sin^4 heta + Y^2cos^2 hetaleft(1 - X ight)^2 ight)^{1/2

Quasi-Longitudinal Propagation in a Collisionless Plasma

If we further assume that the wave propagation is primarily in the direction of the magnetic field, i.e., heta approx 0, we can neglect the Y^4sin^4 heta term above. Thus, for quasi-longitudinal propagation in a cold, collisionless plasma, the Appleton-Hartree equation becomes,

:n^2 = 1 - frac{X}{1 - frac{frac{1}{2}Y^2sin^2 heta}{1 - X} pm Ycos heta}

References

;Citations and notes

ee also

Plasma (physics)


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